Multiple Measurements of a single Qubit

I'm working on my master thesis in the field of quantum theory; currently I investigante No-Go-Theorems like the No-Cloning, No-Deleting, No-Hiding, No-Communication-Theorems ans so on. There is a fundamental question wich is somehow linked to the No-Communication-Theorem.

Is it - in theory - possible to measure one qubits multiple times?
Lets say, you have a qubit |q> = |+> = 0.5^0.5 * |0> + 0.5^0.5 * |1> and perform a measurement in the standard basis. With a probability of 0.5 the outcome of the measurement is |0>. Since the superpostition is destroyed now, the outcome of a second, third, fourth... measurement must be |0> as well. But is it even allowed to perform multiple measurements?
I'm aware that this might be difficult to realize, since photons are destroyed during a measurment. I'm wondering if the theory of quantum mechanics prohibits multiple measurements.

These are my thoughts wich lead to a contradiction to the No-Communication-Theorem, so somewhere must be a mistake:
If I take a qubit |q> = |+> and perform a measurement in the |+>/|-> basis, the outcome would be |+> with probabilty 1. By doing this I fix the qubit in the state |+>. So if it would be possible to perform another measurement in the |0>/|1> basis I would get a random bit. And since |q> is fixed in the state |+>, every following measurement should producea random bit as well.
The No-Communication-Theorem says that it is impossible to communicate via entagled EPR-pairs, especially not faster than light. But if there would be a pair of entagled qubits - one on earth, one on Mars - and the first one is measured in either the |0>/|1> basis or the |+>/|-> basis, then the person on mars would be able to find out wich basis was used by performing multiple measurements in both basises; for one basis he always gets the same result, for the other basis he gets different results.

So... this is known to be impossible. Maybe the answer is just, that not even in theory it is possible to perform multiple measurements on one qubit, but I never heard or read about such a limitation.

The entanglement only applies to the first measurement you do. After that the qubits act independently again, so the |+>-vs-|-> basis measurement will kick the qubit into one of those values and keep returning it even if the other side used |0>-vs-|1> for their measurement.

You can even test whether or not your strategy would work. Go to this blog post, scroll down to the "Write Your Own Quantum Communication Strategy" section, and enter these strategies into the simulator widget:

Alice:
turn(X, 90)
measure()

Bob:
move = measure() != measure()

If Bob's measurements were differing based on what Alice did, you'd start winning the game more than 50% of the time.

@kith: I meant, that after the measurement the qubit is definetly in the state |+> an not in some kind of a superposition. But you are right, its absolutely possible to change the state again, so its not really fixed.

What Strilanc wrote answers my question. I tought the entanglement would continue to exist, but it sounds logically that its not. So the guy on Mars only has a connection to the earth qubit with the first measurement.